LMFDB - Embedded newform 1024.2.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,2,Mod(1,1024)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1024.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.17668116698$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 512)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	1024.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61313 q^{3} -3.41421 q^{5} -1.53073 q^{7} +3.82843 q^{9} +O(q^{10})$	$q+2.61313 q^{3} -3.41421 q^{5} -1.53073 q^{7} +3.82843 q^{9} -4.77791 q^{11} -0.585786 q^{13} -8.92177 q^{15} -2.82843 q^{17} -0.448342 q^{19} -4.00000 q^{21} -5.86030 q^{23} +6.65685 q^{25} +2.16478 q^{27} -4.58579 q^{29} +7.39104 q^{31} -12.4853 q^{33} +5.22625 q^{35} +5.07107 q^{37} -1.53073 q^{39} -4.00000 q^{41} -2.61313 q^{43} -13.0711 q^{45} +7.39104 q^{47} -4.65685 q^{49} -7.39104 q^{51} -7.41421 q^{53} +16.3128 q^{55} -1.17157 q^{57} +2.61313 q^{59} -13.0711 q^{61} -5.86030 q^{63} +2.00000 q^{65} -10.0042 q^{67} -15.3137 q^{69} +11.9832 q^{71} +10.4853 q^{73} +17.3952 q^{75} +7.31371 q^{77} +6.12293 q^{79} -5.82843 q^{81} +3.50981 q^{83} +9.65685 q^{85} -11.9832 q^{87} -0.828427 q^{89} +0.896683 q^{91} +19.3137 q^{93} +1.53073 q^{95} +10.8284 q^{97} -18.2919 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q - 8 * q^5 + 4 * q^9	$ 4 q - 8 q^{5} + 4 q^{9} - 8 q^{13} - 16 q^{21} + 4 q^{25} - 24 q^{29} - 16 q^{33} - 8 q^{37} - 16 q^{41} - 24 q^{45} + 4 q^{49} - 24 q^{53} - 16 q^{57} - 24 q^{61} + 8 q^{65} - 16 q^{69} + 8 q^{73} - 16 q^{77} - 12 q^{81} + 16 q^{85} + 8 q^{89} + 32 q^{93} + 32 q^{97}+O(q^{100}) $ 4 * q - 8 * q^5 + 4 * q^9 - 8 * q^13 - 16 * q^21 + 4 * q^25 - 24 * q^29 - 16 * q^33 - 8 * q^37 - 16 * q^41 - 24 * q^45 + 4 * q^49 - 24 * q^53 - 16 * q^57 - 24 * q^61 + 8 * q^65 - 16 * q^69 + 8 * q^73 - 16 * q^77 - 12 * q^81 + 16 * q^85 + 8 * q^89 + 32 * q^93 + 32 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.61313	1.50869	0.754344	−	0.656479i	$-0.227955\pi$
$3$	2.61313	1.50869	0.754344	+	0.656479i	$0.227955\pi$
$4$	0	0
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	−1.53073	−0.578563	−0.289281	−	0.957244i	$-0.593416\pi$
$7$	−1.53073	−0.578563	−0.289281	+	0.957244i	$0.593416\pi$
$8$	0	0
$9$	3.82843	1.27614
$10$	0	0
$11$	−4.77791	−1.44059	−0.720297	−	0.693666i	$-0.755995\pi$
$11$	−4.77791	−1.44059	−0.720297	+	0.693666i	$0.755995\pi$
$12$	0	0
$13$	−0.585786	−0.162468	−0.0812340	−	0.996695i	$-0.525886\pi$
$13$	−0.585786	−0.162468	−0.0812340	+	0.996695i	$0.525886\pi$
$14$	0	0
$15$	−8.92177	−2.30359
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	−0.448342	−0.102857	−0.0514283	−	0.998677i	$-0.516377\pi$
$19$	−0.448342	−0.102857	−0.0514283	+	0.998677i	$0.516377\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−5.86030	−1.22196	−0.610979	−	0.791647i	$-0.709224\pi$
$23$	−5.86030	−1.22196	−0.610979	+	0.791647i	$0.709224\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	2.16478	0.416613
$28$	0	0
$29$	−4.58579	−0.851559	−0.425780	−	0.904827i	$-0.640000\pi$
$29$	−4.58579	−0.851559	−0.425780	+	0.904827i	$0.640000\pi$
$30$	0	0
$31$	7.39104	1.32747	0.663735	−	0.747968i	$-0.268970\pi$
$31$	7.39104	1.32747	0.663735	+	0.747968i	$0.268970\pi$
$32$	0	0
$33$	−12.4853	−2.17341
$34$	0	0
$35$	5.22625	0.883398
$36$	0	0
$37$	5.07107	0.833678	0.416839	−	0.908980i	$-0.363138\pi$
$37$	5.07107	0.833678	0.416839	+	0.908980i	$0.363138\pi$
$38$	0	0
$39$	−1.53073	−0.245114
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−2.61313	−0.398498	−0.199249	−	0.979949i	$-0.563850\pi$
$43$	−2.61313	−0.398498	−0.199249	+	0.979949i	$0.563850\pi$
$44$	0	0
$45$	−13.0711	−1.94852
$46$	0	0
$47$	7.39104	1.07809	0.539047	−	0.842276i	$-0.318785\pi$
$47$	7.39104	1.07809	0.539047	+	0.842276i	$0.318785\pi$
$48$	0	0
$49$	−4.65685	−0.665265
$50$	0	0
$51$	−7.39104	−1.03495
$52$	0	0
$53$	−7.41421	−1.01842	−0.509210	−	0.860642i	$-0.670062\pi$
$53$	−7.41421	−1.01842	−0.509210	+	0.860642i	$0.670062\pi$
$54$	0	0
$55$	16.3128	2.19962
$56$	0	0
$57$	−1.17157	−0.155179
$58$	0	0
$59$	2.61313	0.340200	0.170100	−	0.985427i	$-0.445591\pi$
$59$	2.61313	0.340200	0.170100	+	0.985427i	$0.445591\pi$
$60$	0	0
$61$	−13.0711	−1.67358	−0.836789	−	0.547525i	$-0.815570\pi$
$61$	−13.0711	−1.67358	−0.836789	+	0.547525i	$0.815570\pi$
$62$	0	0
$63$	−5.86030	−0.738329
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−10.0042	−1.22220	−0.611101	−	0.791552i	$-0.709273\pi$
$67$	−10.0042	−1.22220	−0.611101	+	0.791552i	$0.709273\pi$
$68$	0	0
$69$	−15.3137	−1.84355
$70$	0	0
$71$	11.9832	1.42215	0.711074	−	0.703117i	$-0.248209\pi$
$71$	11.9832	1.42215	0.711074	+	0.703117i	$0.248209\pi$
$72$	0	0
$73$	10.4853	1.22721	0.613605	−	0.789613i	$-0.289719\pi$
$73$	10.4853	1.22721	0.613605	+	0.789613i	$0.289719\pi$
$74$	0	0
$75$	17.3952	2.00862
$76$	0	0
$77$	7.31371	0.833474
$78$	0	0
$79$	6.12293	0.688884	0.344442	−	0.938808i	$-0.388068\pi$
$79$	6.12293	0.688884	0.344442	+	0.938808i	$0.388068\pi$
$80$	0	0
$81$	−5.82843	−0.647603
$82$	0	0
$83$	3.50981	0.385252	0.192626	−	0.981272i	$-0.438300\pi$
$83$	3.50981	0.385252	0.192626	+	0.981272i	$0.438300\pi$
$84$	0	0
$85$	9.65685	1.04743
$86$	0	0
$87$	−11.9832	−1.28474
$88$	0	0
$89$	−0.828427	−0.0878131	−0.0439065	−	0.999036i	$-0.513980\pi$
$89$	−0.828427	−0.0878131	−0.0439065	+	0.999036i	$0.513980\pi$
$90$	0	0
$91$	0.896683	0.0939979
$92$	0	0
$93$	19.3137	2.00274
$94$	0	0
$95$	1.53073	0.157050
$96$	0	0
$97$	10.8284	1.09946	0.549730	−	0.835342i	$-0.314731\pi$
$97$	10.8284	1.09946	0.549730	+	0.835342i	$0.314731\pi$
$98$	0	0
$99$	−18.2919	−1.83840
$100$	0	0
$101$	−2.92893	−0.291440	−0.145720	−	0.989326i	$-0.546550\pi$
$101$	−2.92893	−0.291440	−0.145720	+	0.989326i	$0.546550\pi$
$102$	0	0
$103$	−2.79884	−0.275777	−0.137889	−	0.990448i	$-0.544032\pi$
$103$	−2.79884	−0.275777	−0.137889	+	0.990448i	$0.544032\pi$
$104$	0	0
$105$	13.6569	1.33277
$106$	0	0
$107$	10.0042	0.967139	0.483569	−	0.875306i	$-0.339340\pi$
$107$	10.0042	0.967139	0.483569	+	0.875306i	$0.339340\pi$
$108$	0	0
$109$	−1.07107	−0.102590	−0.0512948	−	0.998684i	$-0.516335\pi$
$109$	−1.07107	−0.102590	−0.0512948	+	0.998684i	$0.516335\pi$
$110$	0	0
$111$	13.2513	1.25776
$112$	0	0
$113$	3.65685	0.344008	0.172004	−	0.985096i	$-0.444976\pi$
$113$	3.65685	0.344008	0.172004	+	0.985096i	$0.444976\pi$
$114$	0	0
$115$	20.0083	1.86579
$116$	0	0
$117$	−2.24264	−0.207332
$118$	0	0
$119$	4.32957	0.396891
$120$	0	0
$121$	11.8284	1.07531
$122$	0	0
$123$	−10.4525	−0.942471
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−13.5140	−1.19917	−0.599586	−	0.800311i	$-0.704668\pi$
$127$	−13.5140	−1.19917	−0.599586	+	0.800311i	$0.704668\pi$
$128$	0	0
$129$	−6.82843	−0.601209
$130$	0	0
$131$	9.10748	0.795724	0.397862	−	0.917445i	$-0.369752\pi$
$131$	9.10748	0.795724	0.397862	+	0.917445i	$0.369752\pi$
$132$	0	0
$133$	0.686292	0.0595090
$134$	0	0
$135$	−7.39104	−0.636119
$136$	0	0
$137$	−20.9706	−1.79164	−0.895818	−	0.444421i	$-0.853409\pi$
$137$	−20.9706	−1.79164	−0.895818	+	0.444421i	$0.853409\pi$
$138$	0	0
$139$	−19.1886	−1.62755	−0.813776	−	0.581178i	$-0.802592\pi$
$139$	−19.1886	−1.62755	−0.813776	+	0.581178i	$0.802592\pi$
$140$	0	0
$141$	19.3137	1.62651
$142$	0	0
$143$	2.79884	0.234050
$144$	0	0
$145$	15.6569	1.30023
$146$	0	0
$147$	−12.1689	−1.00368
$148$	0	0
$149$	−10.2426	−0.839110	−0.419555	−	0.907730i	$-0.637814\pi$
$149$	−10.2426	−0.839110	−0.419555	+	0.907730i	$0.637814\pi$
$150$	0	0
$151$	8.92177	0.726043	0.363022	−	0.931781i	$-0.381745\pi$
$151$	8.92177	0.726043	0.363022	+	0.931781i	$0.381745\pi$
$152$	0	0
$153$	−10.8284	−0.875426
$154$	0	0
$155$	−25.2346	−2.02689
$156$	0	0
$157$	6.24264	0.498217	0.249108	−	0.968476i	$-0.419862\pi$
$157$	6.24264	0.498217	0.249108	+	0.968476i	$0.419862\pi$
$158$	0	0
$159$	−19.3743	−1.53648
$160$	0	0
$161$	8.97056	0.706979
$162$	0	0
$163$	13.0656	1.02338	0.511690	−	0.859170i	$-0.329020\pi$
$163$	13.0656	1.02338	0.511690	+	0.859170i	$0.329020\pi$
$164$	0	0
$165$	42.6274	3.31854
$166$	0	0
$167$	−1.53073	−0.118452	−0.0592259	−	0.998245i	$-0.518863\pi$
$167$	−1.53073	−0.118452	−0.0592259	+	0.998245i	$0.518863\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	−1.71644	−0.131260
$172$	0	0
$173$	−23.2132	−1.76487	−0.882434	−	0.470437i	$-0.844096\pi$
$173$	−23.2132	−1.76487	−0.882434	+	0.470437i	$0.844096\pi$
$174$	0	0
$175$	−10.1899	−0.770282
$176$	0	0
$177$	6.82843	0.513256
$178$	0	0
$179$	1.71644	0.128293	0.0641465	−	0.997940i	$-0.479568\pi$
$179$	1.71644	0.128293	0.0641465	+	0.997940i	$0.479568\pi$
$180$	0	0
$181$	5.75736	0.427941	0.213971	−	0.976840i	$-0.431360\pi$
$181$	5.75736	0.427941	0.213971	+	0.976840i	$0.431360\pi$
$182$	0	0
$183$	−34.1563	−2.52491
$184$	0	0
$185$	−17.3137	−1.27293
$186$	0	0
$187$	13.5140	0.988239
$188$	0	0
$189$	−3.31371	−0.241037
$190$	0	0
$191$	13.5140	0.977837	0.488918	−	0.872330i	$-0.337392\pi$
$191$	13.5140	0.977837	0.488918	+	0.872330i	$0.337392\pi$
$192$	0	0
$193$	−16.4853	−1.18664	−0.593318	−	0.804968i	$-0.702182\pi$
$193$	−16.4853	−1.18664	−0.593318	+	0.804968i	$0.702182\pi$
$194$	0	0
$195$	5.22625	0.374260
$196$	0	0
$197$	−14.7279	−1.04932	−0.524660	−	0.851312i	$-0.675808\pi$
$197$	−14.7279	−1.04932	−0.524660	+	0.851312i	$0.675808\pi$
$198$	0	0
$199$	−22.4357	−1.59043	−0.795214	−	0.606329i	$-0.792641\pi$
$199$	−22.4357	−1.59043	−0.795214	+	0.606329i	$0.792641\pi$
$200$	0	0
$201$	−26.1421	−1.84392
$202$	0	0
$203$	7.01962	0.492681
$204$	0	0
$205$	13.6569	0.953836
$206$	0	0
$207$	−22.4357	−1.55939
$208$	0	0
$209$	2.14214	0.148175
$210$	0	0
$211$	26.9510	1.85538	0.927692	−	0.373346i	$-0.121789\pi$
$211$	26.9510	1.85538	0.927692	+	0.373346i	$0.121789\pi$
$212$	0	0
$213$	31.3137	2.14558
$214$	0	0
$215$	8.92177	0.608460
$216$	0	0
$217$	−11.3137	−0.768025
$218$	0	0
$219$	27.3994	1.85148
$220$	0	0
$221$	1.65685	0.111452
$222$	0	0
$223$	7.39104	0.494940	0.247470	−	0.968896i	$-0.420401\pi$
$223$	7.39104	0.494940	0.247470	+	0.968896i	$0.420401\pi$
$224$	0	0
$225$	25.4853	1.69902
$226$	0	0
$227$	2.61313	0.173439	0.0867196	−	0.996233i	$-0.472362\pi$
$227$	2.61313	0.173439	0.0867196	+	0.996233i	$0.472362\pi$
$228$	0	0
$229$	15.8995	1.05067	0.525334	−	0.850896i	$-0.323940\pi$
$229$	15.8995	1.05067	0.525334	+	0.850896i	$0.323940\pi$
$230$	0	0
$231$	19.1116	1.25745
$232$	0	0
$233$	15.1716	0.993923	0.496961	−	0.867773i	$-0.334449\pi$
$233$	15.1716	0.993923	0.496961	+	0.867773i	$0.334449\pi$
$234$	0	0
$235$	−25.2346	−1.64612
$236$	0	0
$237$	16.0000	1.03931
$238$	0	0
$239$	−29.5641	−1.91235	−0.956173	−	0.292803i	$-0.905412\pi$
$239$	−29.5641	−1.91235	−0.956173	+	0.292803i	$0.905412\pi$
$240$	0	0
$241$	−18.8284	−1.21285	−0.606423	−	0.795142i	$-0.707396\pi$
$241$	−18.8284	−1.21285	−0.606423	+	0.795142i	$0.707396\pi$
$242$	0	0
$243$	−21.7248	−1.39364
$244$	0	0
$245$	15.8995	1.01578
$246$	0	0
$247$	0.262632	0.0167109
$248$	0	0
$249$	9.17157	0.581225
$250$	0	0
$251$	−13.9623	−0.881293	−0.440647	−	0.897681i	$-0.645251\pi$
$251$	−13.9623	−0.881293	−0.440647	+	0.897681i	$0.645251\pi$
$252$	0	0
$253$	28.0000	1.76034
$254$	0	0
$255$	25.2346	1.58025
$256$	0	0
$257$	18.9706	1.18335	0.591676	−	0.806176i	$-0.298467\pi$
$257$	18.9706	1.18335	0.591676	+	0.806176i	$0.298467\pi$
$258$	0	0
$259$	−7.76245	−0.482335
$260$	0	0
$261$	−17.5563	−1.08671
$262$	0	0
$263$	20.6424	1.27286	0.636432	−	0.771333i	$-0.280410\pi$
$263$	20.6424	1.27286	0.636432	+	0.771333i	$0.280410\pi$
$264$	0	0
$265$	25.3137	1.55501
$266$	0	0
$267$	−2.16478	−0.132483
$268$	0	0
$269$	5.55635	0.338777	0.169388	−	0.985549i	$-0.445821\pi$
$269$	5.55635	0.338777	0.169388	+	0.985549i	$0.445821\pi$
$270$	0	0
$271$	28.2960	1.71886	0.859431	−	0.511252i	$-0.170818\pi$
$271$	28.2960	1.71886	0.859431	+	0.511252i	$0.170818\pi$
$272$	0	0
$273$	2.34315	0.141814
$274$	0	0
$275$	−31.8059	−1.91797
$276$	0	0
$277$	0.585786	0.0351965	0.0175982	−	0.999845i	$-0.494398\pi$
$277$	0.585786	0.0351965	0.0175982	+	0.999845i	$0.494398\pi$
$278$	0	0
$279$	28.2960	1.69404
$280$	0	0
$281$	−21.7990	−1.30042	−0.650209	−	0.759755i	$-0.725319\pi$
$281$	−21.7990	−1.30042	−0.650209	+	0.759755i	$0.725319\pi$
$282$	0	0
$283$	9.10748	0.541383	0.270692	−	0.962666i	$-0.412748\pi$
$283$	9.10748	0.541383	0.270692	+	0.962666i	$0.412748\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	6.12293	0.361425
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	28.2960	1.65874
$292$	0	0
$293$	−3.41421	−0.199460	−0.0997302	−	0.995015i	$-0.531798\pi$
$293$	−3.41421	−0.199460	−0.0997302	+	0.995015i	$0.531798\pi$
$294$	0	0
$295$	−8.92177	−0.519446
$296$	0	0
$297$	−10.3431	−0.600170
$298$	0	0
$299$	3.43289	0.198529
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	−7.65367	−0.439692
$304$	0	0
$305$	44.6274	2.55536
$306$	0	0
$307$	−25.6829	−1.46580	−0.732901	−	0.680336i	$-0.761834\pi$
$307$	−25.6829	−1.46580	−0.732901	+	0.680336i	$0.761834\pi$
$308$	0	0
$309$	−7.31371	−0.416062
$310$	0	0
$311$	−7.12840	−0.404215	−0.202107	−	0.979363i	$-0.564779\pi$
$311$	−7.12840	−0.404215	−0.202107	+	0.979363i	$0.564779\pi$
$312$	0	0
$313$	−8.68629	−0.490978	−0.245489	−	0.969399i	$-0.578949\pi$
$313$	−8.68629	−0.490978	−0.245489	+	0.969399i	$0.578949\pi$
$314$	0	0
$315$	20.0083	1.12734
$316$	0	0
$317$	−12.5858	−0.706888	−0.353444	−	0.935456i	$-0.614990\pi$
$317$	−12.5858	−0.706888	−0.353444	+	0.935456i	$0.614990\pi$
$318$	0	0
$319$	21.9105	1.22675
$320$	0	0
$321$	26.1421	1.45911
$322$	0	0
$323$	1.26810	0.0705590
$324$	0	0
$325$	−3.89949	−0.216305
$326$	0	0
$327$	−2.79884	−0.154776
$328$	0	0
$329$	−11.3137	−0.623745
$330$	0	0
$331$	5.67459	0.311904	0.155952	−	0.987765i	$-0.450156\pi$
$331$	5.67459	0.311904	0.155952	+	0.987765i	$0.450156\pi$
$332$	0	0
$333$	19.4142	1.06389
$334$	0	0
$335$	34.1563	1.86616
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	9.55582	0.519001
$340$	0	0
$341$	−35.3137	−1.91234
$342$	0	0
$343$	17.8435	0.963461
$344$	0	0
$345$	52.2843	2.81489
$346$	0	0
$347$	21.7248	1.16625	0.583123	−	0.812384i	$-0.301830\pi$
$347$	21.7248	1.16625	0.583123	+	0.812384i	$0.301830\pi$
$348$	0	0
$349$	22.7279	1.21660	0.608299	−	0.793708i	$-0.291852\pi$
$349$	22.7279	1.21660	0.608299	+	0.793708i	$0.291852\pi$
$350$	0	0
$351$	−1.26810	−0.0676862
$352$	0	0
$353$	−10.9706	−0.583904	−0.291952	−	0.956433i	$-0.594305\pi$
$353$	−10.9706	−0.583904	−0.291952	+	0.956433i	$0.594305\pi$
$354$	0	0
$355$	−40.9133	−2.17145
$356$	0	0
$357$	11.3137	0.598785
$358$	0	0
$359$	−8.92177	−0.470873	−0.235437	−	0.971890i	$-0.575652\pi$
$359$	−8.92177	−0.470873	−0.235437	+	0.971890i	$0.575652\pi$
$360$	0	0
$361$	−18.7990	−0.989421
$362$	0	0
$363$	30.9092	1.62231
$364$	0	0
$365$	−35.7990	−1.87380
$366$	0	0
$367$	−22.1731	−1.15743	−0.578713	−	0.815531i	$-0.696445\pi$
$367$	−22.1731	−1.15743	−0.578713	+	0.815531i	$0.696445\pi$
$368$	0	0
$369$	−15.3137	−0.797200
$370$	0	0
$371$	11.3492	0.589220
$372$	0	0
$373$	−18.7279	−0.969695	−0.484848	−	0.874599i	$-0.661125\pi$
$373$	−18.7279	−0.969695	−0.484848	+	0.874599i	$0.661125\pi$
$374$	0	0
$375$	−14.7821	−0.763343
$376$	0	0
$377$	2.68629	0.138351
$378$	0	0
$379$	4.40649	0.226346	0.113173	−	0.993575i	$-0.463899\pi$
$379$	4.40649	0.226346	0.113173	+	0.993575i	$0.463899\pi$
$380$	0	0
$381$	−35.3137	−1.80918
$382$	0	0
$383$	−8.65914	−0.442461	−0.221231	−	0.975222i	$-0.571007\pi$
$383$	−8.65914	−0.442461	−0.221231	+	0.975222i	$0.571007\pi$
$384$	0	0
$385$	−24.9706	−1.27262
$386$	0	0
$387$	−10.0042	−0.508540
$388$	0	0
$389$	16.3848	0.830741	0.415371	−	0.909652i	$-0.363652\pi$
$389$	16.3848	0.830741	0.415371	+	0.909652i	$0.363652\pi$
$390$	0	0
$391$	16.5754	0.838256
$392$	0	0
$393$	23.7990	1.20050
$394$	0	0
$395$	−20.9050	−1.05185
$396$	0	0
$397$	−9.07107	−0.455264	−0.227632	−	0.973747i	$-0.573098\pi$
$397$	−9.07107	−0.455264	−0.227632	+	0.973747i	$0.573098\pi$
$398$	0	0
$399$	1.79337	0.0897806
$400$	0	0
$401$	24.4853	1.22274	0.611368	−	0.791346i	$-0.290619\pi$
$401$	24.4853	1.22274	0.611368	+	0.791346i	$0.290619\pi$
$402$	0	0
$403$	−4.32957	−0.215671
$404$	0	0
$405$	19.8995	0.988814
$406$	0	0
$407$	−24.2291	−1.20099
$408$	0	0
$409$	−8.68629	−0.429509	−0.214755	−	0.976668i	$-0.568895\pi$
$409$	−8.68629	−0.429509	−0.214755	+	0.976668i	$0.568895\pi$
$410$	0	0
$411$	−54.7987	−2.70302
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−11.9832	−0.588234
$416$	0	0
$417$	−50.1421	−2.45547
$418$	0	0
$419$	−14.3337	−0.700249	−0.350124	−	0.936703i	$-0.613861\pi$
$419$	−14.3337	−0.700249	−0.350124	+	0.936703i	$0.613861\pi$
$420$	0	0
$421$	−11.4142	−0.556295	−0.278147	−	0.960538i	$-0.589720\pi$
$421$	−11.4142	−0.556295	−0.278147	+	0.960538i	$0.589720\pi$
$422$	0	0
$423$	28.2960	1.37580
$424$	0	0
$425$	−18.8284	−0.913313
$426$	0	0
$427$	20.0083	0.968271
$428$	0	0
$429$	7.31371	0.353109
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−25.4558	−1.22333	−0.611665	−	0.791117i	$-0.709500\pi$
$433$	−25.4558	−1.22333	−0.611665	+	0.791117i	$0.709500\pi$
$434$	0	0
$435$	40.9133	1.96164
$436$	0	0
$437$	2.62742	0.125686
$438$	0	0
$439$	18.8490	0.899614	0.449807	−	0.893126i	$-0.351493\pi$
$439$	18.8490	0.899614	0.449807	+	0.893126i	$0.351493\pi$
$440$	0	0
$441$	−17.8284	−0.848973
$442$	0	0
$443$	−10.0042	−0.475312	−0.237656	−	0.971349i	$-0.576379\pi$
$443$	−10.0042	−0.475312	−0.237656	+	0.971349i	$0.576379\pi$
$444$	0	0
$445$	2.82843	0.134080
$446$	0	0
$447$	−26.7653	−1.26596
$448$	0	0
$449$	22.1421	1.04495	0.522476	−	0.852654i	$-0.325008\pi$
$449$	22.1421	1.04495	0.522476	+	0.852654i	$0.325008\pi$
$450$	0	0
$451$	19.1116	0.899932
$452$	0	0
$453$	23.3137	1.09537
$454$	0	0
$455$	−3.06147	−0.143524
$456$	0	0
$457$	18.6274	0.871354	0.435677	−	0.900103i	$-0.356509\pi$
$457$	18.6274	0.871354	0.435677	+	0.900103i	$0.356509\pi$
$458$	0	0
$459$	−6.12293	−0.285794
$460$	0	0
$461$	−29.7574	−1.38594	−0.692969	−	0.720967i	$-0.743698\pi$
$461$	−29.7574	−1.38594	−0.692969	+	0.720967i	$0.743698\pi$
$462$	0	0
$463$	2.53620	0.117867	0.0589337	−	0.998262i	$-0.481230\pi$
$463$	2.53620	0.117867	0.0589337	+	0.998262i	$0.481230\pi$
$464$	0	0
$465$	−65.9411	−3.05795
$466$	0	0
$467$	17.9205	0.829260	0.414630	−	0.909990i	$-0.363911\pi$
$467$	17.9205	0.829260	0.414630	+	0.909990i	$0.363911\pi$
$468$	0	0
$469$	15.3137	0.707121
$470$	0	0
$471$	16.3128	0.751654
$472$	0	0
$473$	12.4853	0.574074
$474$	0	0
$475$	−2.98454	−0.136940
$476$	0	0
$477$	−28.3848	−1.29965
$478$	0	0
$479$	−40.5419	−1.85241	−0.926204	−	0.377024i	$-0.876948\pi$
$479$	−40.5419	−1.85241	−0.926204	+	0.377024i	$0.876948\pi$
$480$	0	0
$481$	−2.97056	−0.135446
$482$	0	0
$483$	23.4412	1.06661
$484$	0	0
$485$	−36.9706	−1.67875
$486$	0	0
$487$	−37.2178	−1.68650	−0.843250	−	0.537521i	$-0.819361\pi$
$487$	−37.2178	−1.68650	−0.843250	+	0.537521i	$0.819361\pi$
$488$	0	0
$489$	34.1421	1.54396
$490$	0	0
$491$	−17.3952	−0.785034	−0.392517	−	0.919745i	$-0.628396\pi$
$491$	−17.3952	−0.785034	−0.392517	+	0.919745i	$0.628396\pi$
$492$	0	0
$493$	12.9706	0.584165
$494$	0	0
$495$	62.4524	2.80703
$496$	0	0
$497$	−18.3431	−0.822803
$498$	0	0
$499$	16.4985	0.738575	0.369287	−	0.929315i	$-0.379602\pi$
$499$	16.4985	0.738575	0.369287	+	0.929315i	$0.379602\pi$
$500$	0	0
$501$	−4.00000	−0.178707
$502$	0	0
$503$	−18.1062	−0.807314	−0.403657	−	0.914910i	$-0.632261\pi$
$503$	−18.1062	−0.807314	−0.403657	+	0.914910i	$0.632261\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	−33.0740	−1.46887
$508$	0	0
$509$	−17.7574	−0.787081	−0.393541	−	0.919307i	$-0.628750\pi$
$509$	−17.7574	−0.787081	−0.393541	+	0.919307i	$0.628750\pi$
$510$	0	0
$511$	−16.0502	−0.710018
$512$	0	0
$513$	−0.970563	−0.0428514
$514$	0	0
$515$	9.55582	0.421080
$516$	0	0
$517$	−35.3137	−1.55310
$518$	0	0
$519$	−60.6590	−2.66264
$520$	0	0
$521$	17.6569	0.773561	0.386780	−	0.922172i	$-0.373587\pi$
$521$	17.6569	0.773561	0.386780	+	0.922172i	$0.373587\pi$
$522$	0	0
$523$	39.5683	1.73020	0.865101	−	0.501598i	$-0.167254\pi$
$523$	39.5683	1.73020	0.865101	+	0.501598i	$0.167254\pi$
$524$	0	0
$525$	−26.6274	−1.16212
$526$	0	0
$527$	−20.9050	−0.910636
$528$	0	0
$529$	11.3431	0.493180
$530$	0	0
$531$	10.0042	0.434144
$532$	0	0
$533$	2.34315	0.101493
$534$	0	0
$535$	−34.1563	−1.47671
$536$	0	0
$537$	4.48528	0.193554
$538$	0	0
$539$	22.2500	0.958377
$540$	0	0
$541$	−1.75736	−0.0755548	−0.0377774	−	0.999286i	$-0.512028\pi$
$541$	−1.75736	−0.0755548	−0.0377774	+	0.999286i	$0.512028\pi$
$542$	0	0
$543$	15.0447	0.645630
$544$	0	0
$545$	3.65685	0.156642
$546$	0	0
$547$	−1.34502	−0.0575091	−0.0287545	−	0.999587i	$-0.509154\pi$
$547$	−1.34502	−0.0575091	−0.0287545	+	0.999587i	$0.509154\pi$
$548$	0	0
$549$	−50.0416	−2.13572
$550$	0	0
$551$	2.05600	0.0875885
$552$	0	0
$553$	−9.37258	−0.398563
$554$	0	0
$555$	−45.2429	−1.92045
$556$	0	0
$557$	−8.58579	−0.363791	−0.181896	−	0.983318i	$-0.558223\pi$
$557$	−8.58579	−0.363791	−0.181896	+	0.983318i	$0.558223\pi$
$558$	0	0
$559$	1.53073	0.0647431
$560$	0	0
$561$	35.3137	1.49095
$562$	0	0
$563$	−21.3533	−0.899936	−0.449968	−	0.893045i	$-0.648565\pi$
$563$	−21.3533	−0.899936	−0.449968	+	0.893045i	$0.648565\pi$
$564$	0	0
$565$	−12.4853	−0.525260
$566$	0	0
$567$	8.92177	0.374679
$568$	0	0
$569$	−14.3431	−0.601296	−0.300648	−	0.953735i	$-0.597203\pi$
$569$	−14.3431	−0.601296	−0.300648	+	0.953735i	$0.597203\pi$
$570$	0	0
$571$	−22.2500	−0.931135	−0.465567	−	0.885012i	$-0.654150\pi$
$571$	−22.2500	−0.931135	−0.465567	+	0.885012i	$0.654150\pi$
$572$	0	0
$573$	35.3137	1.47525
$574$	0	0
$575$	−39.0112	−1.62688
$576$	0	0
$577$	−5.31371	−0.221213	−0.110606	−	0.993864i	$-0.535279\pi$
$577$	−5.31371	−0.221213	−0.110606	+	0.993864i	$0.535279\pi$
$578$	0	0
$579$	−43.0781	−1.79027
$580$	0	0
$581$	−5.37258	−0.222892
$582$	0	0
$583$	35.4244	1.46713
$584$	0	0
$585$	7.65685	0.316572
$586$	0	0
$587$	29.1158	1.20174	0.600869	−	0.799348i	$-0.294821\pi$
$587$	29.1158	1.20174	0.600869	+	0.799348i	$0.294821\pi$
$588$	0	0
$589$	−3.31371	−0.136539
$590$	0	0
$591$	−38.4859	−1.58310
$592$	0	0
$593$	−13.3137	−0.546728	−0.273364	−	0.961911i	$-0.588136\pi$
$593$	−13.3137	−0.546728	−0.273364	+	0.961911i	$0.588136\pi$
$594$	0	0
$595$	−14.7821	−0.606006
$596$	0	0
$597$	−58.6274	−2.39946
$598$	0	0
$599$	−15.7875	−0.645061	−0.322531	−	0.946559i	$-0.604534\pi$
$599$	−15.7875	−0.645061	−0.322531	+	0.946559i	$0.604534\pi$
$600$	0	0
$601$	15.1716	0.618861	0.309431	−	0.950922i	$-0.399862\pi$
$601$	15.1716	0.618861	0.309431	+	0.950922i	$0.399862\pi$
$602$	0	0
$603$	−38.3002	−1.55970
$604$	0	0
$605$	−40.3848	−1.64187
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	18.3431	0.743302
$610$	0	0
$611$	−4.32957	−0.175156
$612$	0	0
$613$	36.5858	1.47769	0.738843	−	0.673878i	$-0.235373\pi$
$613$	36.5858	1.47769	0.738843	+	0.673878i	$0.235373\pi$
$614$	0	0
$615$	35.6871	1.43904
$616$	0	0
$617$	21.5147	0.866150	0.433075	−	0.901358i	$-0.357429\pi$
$617$	21.5147	0.866150	0.433075	+	0.901358i	$0.357429\pi$
$618$	0	0
$619$	−4.77791	−0.192040	−0.0960202	−	0.995379i	$-0.530611\pi$
$619$	−4.77791	−0.192040	−0.0960202	+	0.995379i	$0.530611\pi$
$620$	0	0
$621$	−12.6863	−0.509083
$622$	0	0
$623$	1.26810	0.0508054
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	5.59767	0.223549
$628$	0	0
$629$	−14.3431	−0.571899
$630$	0	0
$631$	2.79884	0.111420	0.0557099	−	0.998447i	$-0.482258\pi$
$631$	2.79884	0.111420	0.0557099	+	0.998447i	$0.482258\pi$
$632$	0	0
$633$	70.4264	2.79920
$634$	0	0
$635$	46.1396	1.83099
$636$	0	0
$637$	2.72792	0.108084
$638$	0	0
$639$	45.8770	1.81486
$640$	0	0
$641$	−9.85786	−0.389362	−0.194681	−	0.980867i	$-0.562367\pi$
$641$	−9.85786	−0.389362	−0.194681	+	0.980867i	$0.562367\pi$
$642$	0	0
$643$	−1.34502	−0.0530426	−0.0265213	−	0.999648i	$-0.508443\pi$
$643$	−1.34502	−0.0530426	−0.0265213	+	0.999648i	$0.508443\pi$
$644$	0	0
$645$	23.3137	0.917976
$646$	0	0
$647$	−0.262632	−0.0103251	−0.00516257	−	0.999987i	$-0.501643\pi$
$647$	−0.262632	−0.0103251	−0.00516257	+	0.999987i	$0.501643\pi$
$648$	0	0
$649$	−12.4853	−0.490090
$650$	0	0
$651$	−29.5641	−1.15871
$652$	0	0
$653$	37.5563	1.46969	0.734847	−	0.678233i	$-0.237254\pi$
$653$	37.5563	1.46969	0.734847	+	0.678233i	$0.237254\pi$
$654$	0	0
$655$	−31.0949	−1.21498
$656$	0	0
$657$	40.1421	1.56609
$658$	0	0
$659$	33.0740	1.28838	0.644189	−	0.764866i	$-0.277195\pi$
$659$	33.0740	1.28838	0.644189	+	0.764866i	$0.277195\pi$
$660$	0	0
$661$	−14.9289	−0.580668	−0.290334	−	0.956925i	$-0.593766\pi$
$661$	−14.9289	−0.580668	−0.290334	+	0.956925i	$0.593766\pi$
$662$	0	0
$663$	4.32957	0.168147
$664$	0	0
$665$	−2.34315	−0.0908633
$666$	0	0
$667$	26.8741	1.04057
$668$	0	0
$669$	19.3137	0.746711
$670$	0	0
$671$	62.4524	2.41095
$672$	0	0
$673$	−21.1716	−0.816104	−0.408052	−	0.912959i	$-0.633792\pi$
$673$	−21.1716	−0.816104	−0.408052	+	0.912959i	$0.633792\pi$
$674$	0	0
$675$	14.4107	0.554666
$676$	0	0
$677$	−17.5563	−0.674745	−0.337373	−	0.941371i	$-0.609538\pi$
$677$	−17.5563	−0.674745	−0.337373	+	0.941371i	$0.609538\pi$
$678$	0	0
$679$	−16.5754	−0.636107
$680$	0	0
$681$	6.82843	0.261666
$682$	0	0
$683$	20.0852	0.768541	0.384270	−	0.923221i	$-0.374453\pi$
$683$	20.0852	0.768541	0.384270	+	0.923221i	$0.374453\pi$
$684$	0	0
$685$	71.5980	2.73562
$686$	0	0
$687$	41.5474	1.58513
$688$	0	0
$689$	4.34315	0.165461
$690$	0	0
$691$	−19.9314	−0.758226	−0.379113	−	0.925350i	$-0.623771\pi$
$691$	−19.9314	−0.758226	−0.379113	+	0.925350i	$0.623771\pi$
$692$	0	0
$693$	28.0000	1.06363
$694$	0	0
$695$	65.5139	2.48508
$696$	0	0
$697$	11.3137	0.428537
$698$	0	0
$699$	39.6452	1.49952
$700$	0	0
$701$	28.8701	1.09041	0.545204	−	0.838304i	$-0.316452\pi$
$701$	28.8701	1.09041	0.545204	+	0.838304i	$0.316452\pi$
$702$	0	0
$703$	−2.27357	−0.0857493
$704$	0	0
$705$	−65.9411	−2.48349
$706$	0	0
$707$	4.48342	0.168616
$708$	0	0
$709$	35.6985	1.34068	0.670342	−	0.742052i	$-0.266147\pi$
$709$	35.6985	1.34068	0.670342	+	0.742052i	$0.266147\pi$
$710$	0	0
$711$	23.4412	0.879114
$712$	0	0
$713$	−43.3137	−1.62211
$714$	0	0
$715$	−9.55582	−0.357367
$716$	0	0
$717$	−77.2548	−2.88513
$718$	0	0
$719$	12.2459	0.456694	0.228347	−	0.973580i	$-0.426668\pi$
$719$	12.2459	0.456694	0.228347	+	0.973580i	$0.426668\pi$
$720$	0	0
$721$	4.28427	0.159555
$722$	0	0
$723$	−49.2011	−1.82981
$724$	0	0
$725$	−30.5269	−1.13374
$726$	0	0
$727$	−42.8155	−1.58794	−0.793969	−	0.607958i	$-0.791989\pi$
$727$	−42.8155	−1.58794	−0.793969	+	0.607958i	$0.791989\pi$
$728$	0	0
$729$	−39.2843	−1.45497
$730$	0	0
$731$	7.39104	0.273367
$732$	0	0
$733$	−46.5269	−1.71851	−0.859255	−	0.511547i	$-0.829073\pi$
$733$	−46.5269	−1.71851	−0.859255	+	0.511547i	$0.829073\pi$
$734$	0	0
$735$	41.5474	1.53250
$736$	0	0
$737$	47.7990	1.76070
$738$	0	0
$739$	31.4344	1.15633	0.578167	−	0.815918i	$-0.303768\pi$
$739$	31.4344	1.15633	0.578167	+	0.815918i	$0.303768\pi$
$740$	0	0
$741$	0.686292	0.0252115
$742$	0	0
$743$	−28.5587	−1.04772	−0.523858	−	0.851806i	$-0.675508\pi$
$743$	−28.5587	−1.04772	−0.523858	+	0.851806i	$0.675508\pi$
$744$	0	0
$745$	34.9706	1.28122
$746$	0	0
$747$	13.4370	0.491636
$748$	0	0
$749$	−15.3137	−0.559551
$750$	0	0
$751$	−6.12293	−0.223429	−0.111715	−	0.993740i	$-0.535634\pi$
$751$	−6.12293	−0.223429	−0.111715	+	0.993740i	$0.535634\pi$
$752$	0	0
$753$	−36.4853	−1.32960
$754$	0	0
$755$	−30.4608	−1.10858
$756$	0	0
$757$	33.0711	1.20199	0.600994	−	0.799253i	$-0.294771\pi$
$757$	33.0711	1.20199	0.600994	+	0.799253i	$0.294771\pi$
$758$	0	0
$759$	73.1675	2.65581
$760$	0	0
$761$	−10.6274	−0.385244	−0.192622	−	0.981273i	$-0.561699\pi$
$761$	−10.6274	−0.385244	−0.192622	+	0.981273i	$0.561699\pi$
$762$	0	0
$763$	1.63952	0.0593546
$764$	0	0
$765$	36.9706	1.33667
$766$	0	0
$767$	−1.53073	−0.0552716
$768$	0	0
$769$	10.8284	0.390483	0.195242	−	0.980755i	$-0.437451\pi$
$769$	10.8284	0.390483	0.195242	+	0.980755i	$0.437451\pi$
$770$	0	0
$771$	49.5725	1.78531
$772$	0	0
$773$	19.6985	0.708505	0.354253	−	0.935150i	$-0.384735\pi$
$773$	19.6985	0.708505	0.354253	+	0.935150i	$0.384735\pi$
$774$	0	0
$775$	49.2011	1.76735
$776$	0	0
$777$	−20.2843	−0.727694
$778$	0	0
$779$	1.79337	0.0642540
$780$	0	0
$781$	−57.2548	−2.04874
$782$	0	0
$783$	−9.92724	−0.354771
$784$	0	0
$785$	−21.3137	−0.760719
$786$	0	0
$787$	−48.7527	−1.73785	−0.868923	−	0.494947i	$-0.835187\pi$
$787$	−48.7527	−1.73785	−0.868923	+	0.494947i	$0.835187\pi$
$788$	0	0
$789$	53.9411	1.92035
$790$	0	0
$791$	−5.59767	−0.199030
$792$	0	0
$793$	7.65685	0.271903
$794$	0	0
$795$	66.1479	2.34602
$796$	0	0
$797$	−19.6985	−0.697756	−0.348878	−	0.937168i	$-0.613437\pi$
$797$	−19.6985	−0.697756	−0.348878	+	0.937168i	$0.613437\pi$
$798$	0	0
$799$	−20.9050	−0.739566
$800$	0	0
$801$	−3.17157	−0.112062
$802$	0	0
$803$	−50.0977	−1.76791
$804$	0	0
$805$	−30.6274	−1.07947
$806$	0	0
$807$	14.5194	0.511108
$808$	0	0
$809$	−47.3137	−1.66346	−0.831731	−	0.555179i	$-0.812650\pi$
$809$	−47.3137	−1.66346	−0.831731	+	0.555179i	$0.812650\pi$
$810$	0	0
$811$	20.4567	0.718331	0.359165	−	0.933274i	$-0.383061\pi$
$811$	20.4567	0.718331	0.359165	+	0.933274i	$0.383061\pi$
$812$	0	0
$813$	73.9411	2.59323
$814$	0	0
$815$	−44.6088	−1.56258
$816$	0	0
$817$	1.17157	0.0409881
$818$	0	0
$819$	3.43289	0.119955
$820$	0	0
$821$	26.5269	0.925796	0.462898	−	0.886412i	$-0.346810\pi$
$821$	26.5269	0.925796	0.462898	+	0.886412i	$0.346810\pi$
$822$	0	0
$823$	−42.8155	−1.49245	−0.746227	−	0.665692i	$-0.768137\pi$
$823$	−42.8155	−1.49245	−0.746227	+	0.665692i	$0.768137\pi$
$824$	0	0
$825$	−83.1127	−2.89361
$826$	0	0
$827$	−29.1158	−1.01246	−0.506228	−	0.862400i	$-0.668961\pi$
$827$	−29.1158	−1.01246	−0.506228	+	0.862400i	$0.668961\pi$
$828$	0	0
$829$	−24.3848	−0.846918	−0.423459	−	0.905915i	$-0.639184\pi$
$829$	−24.3848	−0.846918	−0.423459	+	0.905915i	$0.639184\pi$
$830$	0	0
$831$	1.53073	0.0531006
$832$	0	0
$833$	13.1716	0.456368
$834$	0	0
$835$	5.22625	0.180862
$836$	0	0
$837$	16.0000	0.553041
$838$	0	0
$839$	−39.7540	−1.37246	−0.686231	−	0.727384i	$-0.740736\pi$
$839$	−39.7540	−1.37246	−0.686231	+	0.727384i	$0.740736\pi$
$840$	0	0
$841$	−7.97056	−0.274847
$842$	0	0
$843$	−56.9635	−1.96193
$844$	0	0
$845$	43.2132	1.48658
$846$	0	0
$847$	−18.1062	−0.622135
$848$	0	0
$849$	23.7990	0.816779
$850$	0	0
$851$	−29.7180	−1.01872
$852$	0	0
$853$	51.8995	1.77700	0.888502	−	0.458872i	$-0.151746\pi$
$853$	51.8995	1.77700	0.888502	+	0.458872i	$0.151746\pi$
$854$	0	0
$855$	5.86030	0.200418
$856$	0	0
$857$	−32.2843	−1.10281	−0.551405	−	0.834238i	$-0.685908\pi$
$857$	−32.2843	−1.10281	−0.551405	+	0.834238i	$0.685908\pi$
$858$	0	0
$859$	−28.7444	−0.980746	−0.490373	−	0.871513i	$-0.663139\pi$
$859$	−28.7444	−0.980746	−0.490373	+	0.871513i	$0.663139\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	0	0
$863$	43.0781	1.46640	0.733198	−	0.680015i	$-0.238027\pi$
$863$	43.0781	1.46640	0.733198	+	0.680015i	$0.238027\pi$
$864$	0	0
$865$	79.2548	2.69475
$866$	0	0
$867$	−23.5181	−0.798718
$868$	0	0
$869$	−29.2548	−0.992402
$870$	0	0
$871$	5.86030	0.198569
$872$	0	0
$873$	41.4558	1.40307
$874$	0	0
$875$	8.65914	0.292732
$876$	0	0
$877$	−18.4437	−0.622798	−0.311399	−	0.950279i	$-0.600798\pi$
$877$	−18.4437	−0.622798	−0.311399	+	0.950279i	$0.600798\pi$
$878$	0	0
$879$	−8.92177	−0.300924
$880$	0	0
$881$	27.9411	0.941360	0.470680	−	0.882304i	$-0.344009\pi$
$881$	27.9411	0.941360	0.470680	+	0.882304i	$0.344009\pi$
$882$	0	0
$883$	−2.98454	−0.100438	−0.0502190	−	0.998738i	$-0.515992\pi$
$883$	−2.98454	−0.100438	−0.0502190	+	0.998738i	$0.515992\pi$
$884$	0	0
$885$	−23.3137	−0.783682
$886$	0	0
$887$	10.1899	0.342142	0.171071	−	0.985259i	$-0.445277\pi$
$887$	10.1899	0.342142	0.171071	+	0.985259i	$0.445277\pi$
$888$	0	0
$889$	20.6863	0.693796
$890$	0	0
$891$	27.8477	0.932933
$892$	0	0
$893$	−3.31371	−0.110889
$894$	0	0
$895$	−5.86030	−0.195888
$896$	0	0
$897$	8.97056	0.299518
$898$	0	0
$899$	−33.8937	−1.13042
$900$	0	0
$901$	20.9706	0.698631
$902$	0	0
$903$	10.4525	0.347838
$904$	0	0
$905$	−19.6569	−0.653416
$906$	0	0
$907$	−9.47890	−0.314742	−0.157371	−	0.987540i	$-0.550302\pi$
$907$	−9.47890	−0.314742	−0.157371	+	0.987540i	$0.550302\pi$
$908$	0	0
$909$	−11.2132	−0.371918
$910$	0	0
$911$	16.0502	0.531766	0.265883	−	0.964005i	$-0.414337\pi$
$911$	16.0502	0.531766	0.265883	+	0.964005i	$0.414337\pi$
$912$	0	0
$913$	−16.7696	−0.554991
$914$	0	0
$915$	116.617	3.85524
$916$	0	0
$917$	−13.9411	−0.460377
$918$	0	0
$919$	−35.4244	−1.16854	−0.584272	−	0.811558i	$-0.698620\pi$
$919$	−35.4244	−1.16854	−0.584272	+	0.811558i	$0.698620\pi$
$920$	0	0
$921$	−67.1127	−2.21144
$922$	0	0
$923$	−7.01962	−0.231054
$924$	0	0
$925$	33.7574	1.10994
$926$	0	0
$927$	−10.7151	−0.351931
$928$	0	0
$929$	−37.1716	−1.21956	−0.609780	−	0.792571i	$-0.708742\pi$
$929$	−37.1716	−1.21956	−0.609780	+	0.792571i	$0.708742\pi$
$930$	0	0
$931$	2.08786	0.0684269
$932$	0	0
$933$	−18.6274	−0.609834
$934$	0	0
$935$	−46.1396	−1.50893
$936$	0	0
$937$	−7.45584	−0.243572	−0.121786	−	0.992556i	$-0.538862\pi$
$937$	−7.45584	−0.243572	−0.121786	+	0.992556i	$0.538862\pi$
$938$	0	0
$939$	−22.6984	−0.740733
$940$	0	0
$941$	−17.0711	−0.556501	−0.278250	−	0.960509i	$-0.589755\pi$
$941$	−17.0711	−0.556501	−0.278250	+	0.960509i	$0.589755\pi$
$942$	0	0
$943$	23.4412	0.763351
$944$	0	0
$945$	11.3137	0.368035
$946$	0	0
$947$	37.0321	1.20338	0.601691	−	0.798729i	$-0.294494\pi$
$947$	37.0321	1.20338	0.601691	+	0.798729i	$0.294494\pi$
$948$	0	0
$949$	−6.14214	−0.199382
$950$	0	0
$951$	−32.8882	−1.06647
$952$	0	0
$953$	44.9706	1.45674	0.728370	−	0.685184i	$-0.240278\pi$
$953$	44.9706	1.45674	0.728370	+	0.685184i	$0.240278\pi$
$954$	0	0
$955$	−46.1396	−1.49304
$956$	0	0
$957$	57.2548	1.85079
$958$	0	0
$959$	32.1003	1.03657
$960$	0	0
$961$	23.6274	0.762175
$962$	0	0
$963$	38.3002	1.23421
$964$	0	0
$965$	56.2843	1.81185
$966$	0	0
$967$	4.59220	0.147675	0.0738376	−	0.997270i	$-0.476475\pi$
$967$	4.59220	0.147675	0.0738376	+	0.997270i	$0.476475\pi$
$968$	0	0
$969$	3.31371	0.106452
$970$	0	0
$971$	8.58221	0.275416	0.137708	−	0.990473i	$-0.456026\pi$
$971$	8.58221	0.275416	0.137708	+	0.990473i	$0.456026\pi$
$972$	0	0
$973$	29.3726	0.941642
$974$	0	0
$975$	−10.1899	−0.326337
$976$	0	0
$977$	15.1127	0.483498	0.241749	−	0.970339i	$-0.422279\pi$
$977$	15.1127	0.483498	0.241749	+	0.970339i	$0.422279\pi$
$978$	0	0
$979$	3.95815	0.126503
$980$	0	0
$981$	−4.10051	−0.130919
$982$	0	0
$983$	−2.05600	−0.0655762	−0.0327881	−	0.999462i	$-0.510439\pi$
$983$	−2.05600	−0.0655762	−0.0327881	+	0.999462i	$0.510439\pi$
$984$	0	0
$985$	50.2843	1.60219
$986$	0	0
$987$	−29.5641	−0.941037
$988$	0	0
$989$	15.3137	0.486948
$990$	0	0
$991$	−23.4412	−0.744635	−0.372317	−	0.928106i	$-0.621437\pi$
$991$	−23.4412	−0.744635	−0.372317	+	0.928106i	$0.621437\pi$
$992$	0	0
$993$	14.8284	0.470566
$994$	0	0
$995$	76.6004	2.42840
$996$	0	0
$997$	8.38478	0.265549	0.132774	−	0.991146i	$-0.457611\pi$
$997$	8.38478	0.265549	0.132774	+	0.991146i	$0.457611\pi$
$998$	0	0
$999$	10.9778	0.347321

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.a.h.1.4		4
3.2	odd	2		9216.2.a.bp.1.3		4
4.3	odd	2	inner	1024.2.a.h.1.1		4
8.3	odd	2		1024.2.a.i.1.4		4
8.5	even	2		1024.2.a.i.1.1		4
12.11	even	2		9216.2.a.bp.1.4		4
16.3	odd	4		1024.2.b.g.513.2		8
16.5	even	4		1024.2.b.g.513.1		8
16.11	odd	4		1024.2.b.g.513.7		8
16.13	even	4		1024.2.b.g.513.8		8
24.5	odd	2		9216.2.a.w.1.1		4
24.11	even	2		9216.2.a.w.1.2		4
32.3	odd	8		512.2.e.j.129.1	yes	8
32.5	even	8		512.2.e.i.385.1	yes	8
32.11	odd	8		512.2.e.j.385.1	yes	8
32.13	even	8		512.2.e.i.129.1	✓	8
32.19	odd	8		512.2.e.i.129.4	yes	8
32.21	even	8		512.2.e.j.385.4	yes	8
32.27	odd	8		512.2.e.i.385.4	yes	8
32.29	even	8		512.2.e.j.129.4	yes	8
96.5	odd	8		4608.2.k.bi.3457.3		8
96.11	even	8		4608.2.k.bd.3457.2		8
96.29	odd	8		4608.2.k.bd.1153.2		8
96.35	even	8		4608.2.k.bd.1153.1		8
96.53	odd	8		4608.2.k.bd.3457.1		8
96.59	even	8		4608.2.k.bi.3457.4		8
96.77	odd	8		4608.2.k.bi.1153.4		8
96.83	even	8		4608.2.k.bi.1153.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.i.129.1	✓	8	32.13	even	8
512.2.e.i.129.4	yes	8	32.19	odd	8
512.2.e.i.385.1	yes	8	32.5	even	8
512.2.e.i.385.4	yes	8	32.27	odd	8
512.2.e.j.129.1	yes	8	32.3	odd	8
512.2.e.j.129.4	yes	8	32.29	even	8
512.2.e.j.385.1	yes	8	32.11	odd	8
512.2.e.j.385.4	yes	8	32.21	even	8
1024.2.a.h.1.1		4	4.3	odd	2	inner
1024.2.a.h.1.4		4	1.1	even	1	trivial
1024.2.a.i.1.1		4	8.5	even	2
1024.2.a.i.1.4		4	8.3	odd	2
1024.2.b.g.513.1		8	16.5	even	4
1024.2.b.g.513.2		8	16.3	odd	4
1024.2.b.g.513.7		8	16.11	odd	4
1024.2.b.g.513.8		8	16.13	even	4
4608.2.k.bd.1153.1		8	96.35	even	8
4608.2.k.bd.1153.2		8	96.29	odd	8
4608.2.k.bd.3457.1		8	96.53	odd	8
4608.2.k.bd.3457.2		8	96.11	even	8
4608.2.k.bi.1153.3		8	96.83	even	8
4608.2.k.bi.1153.4		8	96.77	odd	8
4608.2.k.bi.3457.3		8	96.5	odd	8
4608.2.k.bi.3457.4		8	96.59	even	8
9216.2.a.w.1.1		4	24.5	odd	2
9216.2.a.w.1.2		4	24.11	even	2
9216.2.a.bp.1.3		4	3.2	odd	2
9216.2.a.bp.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61313 q^{3} -3.41421 q^{5} -1.53073 q^{7} +3.82843 q^{9} +O(q^{10})\)	\(q+2.61313 q^{3} -3.41421 q^{5} -1.53073 q^{7} +3.82843 q^{9} -4.77791 q^{11} -0.585786 q^{13} -8.92177 q^{15} -2.82843 q^{17} -0.448342 q^{19} -4.00000 q^{21} -5.86030 q^{23} +6.65685 q^{25} +2.16478 q^{27} -4.58579 q^{29} +7.39104 q^{31} -12.4853 q^{33} +5.22625 q^{35} +5.07107 q^{37} -1.53073 q^{39} -4.00000 q^{41} -2.61313 q^{43} -13.0711 q^{45} +7.39104 q^{47} -4.65685 q^{49} -7.39104 q^{51} -7.41421 q^{53} +16.3128 q^{55} -1.17157 q^{57} +2.61313 q^{59} -13.0711 q^{61} -5.86030 q^{63} +2.00000 q^{65} -10.0042 q^{67} -15.3137 q^{69} +11.9832 q^{71} +10.4853 q^{73} +17.3952 q^{75} +7.31371 q^{77} +6.12293 q^{79} -5.82843 q^{81} +3.50981 q^{83} +9.65685 q^{85} -11.9832 q^{87} -0.828427 q^{89} +0.896683 q^{91} +19.3137 q^{93} +1.53073 q^{95} +10.8284 q^{97} -18.2919 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q - 8 * q^5 + 4 * q^9	\( 4 q - 8 q^{5} + 4 q^{9} - 8 q^{13} - 16 q^{21} + 4 q^{25} - 24 q^{29} - 16 q^{33} - 8 q^{37} - 16 q^{41} - 24 q^{45} + 4 q^{49} - 24 q^{53} - 16 q^{57} - 24 q^{61} + 8 q^{65} - 16 q^{69} + 8 q^{73} - 16 q^{77} - 12 q^{81} + 16 q^{85} + 8 q^{89} + 32 q^{93} + 32 q^{97}+O(q^{100}) \) 4 * q - 8 * q^5 + 4 * q^9 - 8 * q^13 - 16 * q^21 + 4 * q^25 - 24 * q^29 - 16 * q^33 - 8 * q^37 - 16 * q^41 - 24 * q^45 + 4 * q^49 - 24 * q^53 - 16 * q^57 - 24 * q^61 + 8 * q^65 - 16 * q^69 + 8 * q^73 - 16 * q^77 - 12 * q^81 + 16 * q^85 + 8 * q^89 + 32 * q^93 + 32 * q^97

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